The Locked Track constraint is a bit tricky to explain, both graphically and textual.
Basically, it is a Track To constraint, but with a locked axis, i.e.
an axis that cannot rotate (change its orientation). Hence,
the owner can only track its target by rotating around this axis,
and unless the target is in the plane perpendicular to the locked axis, and crossing the owner,
this owner cannot really point at its target.

Let us take the best real world equivalent: a compass.
It can rotate to point in the general direction of its target
(the magnetic North, or a neighbor magnet), but it cannot point directly at it,
because it spins like a wheel on an axle.
If a compass is sitting on a table and there is a magnet directly above it,
the compass cannot point to it. If we move the magnet more to one side of the compass,
it still cannot point at the target,
but it can point in the general direction of the target,
and still obey its restrictions of the axle.

When using a Locked Track constraint, you can think of the target as a magnet,
and the owner as a compass.
The Lock axis will function as the axle around which the owner spins,
and the To axis will function as the compass’ needle.
Which axis does what is up to you!

If you have trouble understanding the buttons of this constraint, read the tooltips,
they are pretty good. If you do not know where your object’s axes are,
turn on the Axis button in the Object menu’s Draw panel.
Or, if you are working with bones, turn on the Axes button
in the Armature menu’s Display panel.

This constraint was designed to work cooperatively with the Track To constraint.
If you set the axes buttons right for these two constraints,
Track To can be used to point the axle at a primary target,
and Locked Track can spin the owner around that axle to a secondary target.